# Taylor series.

Definition 10.5.1   Let be a function having derivatives of any order at . The power series      is called the Taylor series of the function at .

If , this series is called the Mac-Laurin series of .

Example 10.5.2 (A couple of MacLaurin series)

• • • • • • • , for .
• Theorem 10.5.3   Let and be two functions having Taylor series expansions and respectively, converging on the interval . Then has as its Taylor series expansion the series and it converges on .

Theorem 10.5.4   Let and be two functions having Taylor series expansions and respectively, converging on the interval . Then has as its Taylor series expansion the product of these series and it converges on .

Example 10.5.5   On the one side we have: On the other side, we have:    Thus, we have:   Theorem 10.5.6   Let and be two functions having Taylor series expansions in an open interval about 0. The Taylor series expansion of is obtained by substituting the Taylor series of into the Taylor series of .    Thus:   